Understanding the linear law and lines of best fit

Straight lines are the friendliest graphs in all of mathematics. If a set of points lies on a straight line, you can read its gradient and its intercept straight off the axes, and those two numbers tell you everything about the relationship.

Curves give you none of that comfort, you cannot eyeball the equation of a curve.

The linear law is the trick that gets you the comfort back. The idea is to rewrite a curved relationship between $x$ and $y$ so that it becomes a straight-line relationship between two new quantities, which we usually call $X$ and $Y$.

The target shape is always the same:

$$Y = mX + c$$

Here $Y$ and $X$ are built out of the original $x$ and $y$, they might be $\log y$, or $\frac{y}{x}$, or $x^{2}$, whatever makes the relationship straight. Once you plot $Y$ against $X$ and get a straight line, its gradient is $m$ and its vertical intercept is $c$.

The job of almost every linear-law question is to identify what $Y$, $X$, $m$ and $c$ stand for, and then use the graph to find the constants hidden in the original equation.

Two forms come up again and again, and both are freed by taking logarithms. Suppose the relationship is a power law, $y = a x^{n}$, where $a$ and $n$ are the unknown constants.

Take $\log$ of both sides and use the log laws:

$$\log y = n \log x + \log a$$

Compare that with $Y = mX + c$. If you plot $\log y$ (as $Y$) against $\log x$ (as $X$), you get a straight line whose gradient is $n$ and whose intercept is $\log a$.

Now the second form, an exponential relationship $y = a b^{x}$:

$$\log y = (\log b)\,x + \log a$$

This time you plot $\log y$ against $x$ itself. The gradient is $\log b$ and the intercept is $\log a$.

Notice the difference: in a power law you take the log of both variables, but in an exponential you take the log of $y$ only. Choosing correctly is the single most important decision in the whole chapter.

Once the straight line is drawn, the constants come from its gradient and intercept. The gradient is found from any two well-separated points that lie on the line, not two of the original data points, which is a distinction that costs marks:

$$m = \frac{Y_{2} - Y_{1}}{X_{2} - X_{1}}$$

Suppose you plotted $\log y$ against $\log x$ for a power law and read a gradient of $2$ and a vertical intercept of $0.5$. Then $n = 2$ directly, and $\log a = 0.5$, so $a = 10^{0.5} = \sqrt{10} \approx 3.16$.

The equation is recovered: $y \approx 3.16\,x^{2}$.

The last step trips people up more than the first. The intercept is almost never the constant itself, it is usually $\log a$, so you must undo the log to get $a$.

Write down what the intercept equals before you write down the constant, and that reversal will not slip through.

Real measured data never sits perfectly on a line. The plotted points scatter a little because of experimental error, so you draw a single straight line of best fit that passes as evenly as possible through the cloud of points.

• Aim for roughly the same number of points above the line as below it.
• Keep the points close to the line overall, the line should follow the trend, not chase any single point.
• Do not force the line through the origin unless the physics or the data genuinely says it should pass through $(0,0)$.

Because the line of best fit is a judgement, the examiner allows a reasonable range of answers for the gradient and intercept read from it. What is not forgiven is reading a value off a shaky freehand line, or using original data points instead of points that actually sit on the line you drew.

Use a ruler, use a sharp pencil, and take your two reading points far apart so a small error in reading has a small effect on the gradient.

Linear-law questions are generous with method marks, but only if your working shows the examiner what each axis stands for and where each number came from.

• Taking the log of only one variable when both are needed, or of both when only one is, decide power law versus exponential first.
• Finding the gradient from two of the plotted data points instead of two points read off the best-fit line.
• Muddling the axes: label clearly which quantity is $Y$ and which is $X$ before plotting anything.

The linear law rewards a particular kind of clear thinking: knowing which manipulation straightens a given relationship, and keeping careful track of what each axis and each constant means. That is exactly the sort of habit that is hard to build alone and quick to build with someone watching your working.

One-to-one, a teacher can see the precise step where the log goes on the wrong variable or the intercept is read as the constant, and fix it before it becomes a pattern. Our teachers are experienced; lessons are online and taught in English, while the SPM papers are set bilingually in Bahasa Melayu and English.

If graph work is a weak spot, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We will not promise a grade, but we can help make the linear law one of the calmer, more mechanical parts of your paper.